The Solution of Weakly Nonlinear Oscillatory Problems with No Damping Using MAPLE
|
|
- Sharon Evans
- 5 years ago
- Views:
Transcription
1 World Applied Sciences Journal (): 64-69, 0 ISSN IDOSI Publications, 0 DOI: 0.589/idosi.wasj The Solution of Weakly Nonlinear Oscillatory Problems with No Damping Using MAPLE N. Hashim, N.S. Amin, A.A. Shariff, F.A. Manaf and R.M. Tahir Mathematics Section, Centre for Foundation Studies in Science, University of Malaya, Kuala Lumpur, Malaysia Department of Mathematics, Faculty of Science, University Technology of Malaysia, Johor, Malaysia Abstract: Weakly nonlinear oscillatory problems in engineering dynamics are frequently presented by nonlinear ordinary differential equations. Very few of such equations have exact solutions, thus the need for approximation techniques. In this study, the classical perturbation method, namely the Lindste-Poincare method (L-P method) is discussed. In most work, the L-P method is used to obtain uniformly valid approximate solutions to second order only, as perturbation procedure usually involves cumbersome algebraic manipulations and calculations which are time-consuming and prone to errors. Therefore, in this paper we manage to get the solution to fifth order by using our own built in Maple package based on Maple 4 and we include the graphical presentations of the results. Key words: Weakly nonlinear oscillator Nonlinear ODE perturbation method Lindste-Poincare method INTRODUCTION The Lindsted-Poincare method was introduced by Swedish Mathematician and astronomer Lindste in 88 and French Mathematician and theoretical Physicist Poincare in 886 [-4]. It involves a single time variable or scale and manages to obtain uniformly valid approximate solutions to nonlinear problems. In this paper, firstly we consider a weakly nonlinear oscillator with no damping described as, In 987, Rand and Armbruster [5] used the computer algebra system MACYSMA in a number of perturbation techniques for systems of coupled oscillators and bifurcation theory. In 996, Heck [6] and in 999 Chin and Nayfeh [7] had implemented L-P method using Maple V Release 4, while in 00, Lopez [8] had implemented L-P method using Maple 8. In all the papers mentioned using the method, only second order approximations were obtained at most. In the next section we will consider the Duffing equation to illustrate the L-P method. d y dy + 0 y = F y,, 0< () The Duffing Equation: Duffing equation is the most common example of weakly nonlinear oscillatory system where is a small positive parameter, often called a with no damping where 0 = and dy/ = 0. [9] also have perturbation quantity and 0 (we take, 0 = ) is an chosen the same example and solve it using homotopy angular frequency, which is often referred to as the perturbation method coupled with Laplace transform and natural frequency. In this case, the system is free of Padé approximants. Let s consider the initial value damping whereby, there is no resisting force, such as air problem for the equation, resistance. The periodic solution to the equation () is assumed to be of the form, d y + y = y, () m m+ y( t, ) = ym () t + O( ), () m= 0 with condition, 0 < t < and 0 <. The initial known as asymptotic expansion. conditions for the above problem, Corresponding Author: N. Hashim, Mathematics Section, Centre for Foundation Studies in Science, University of Malaya, Kuala Lumpur, Malaysia. Tel:
2 and is a constant. We take initial conditions (4a,b) for y = sin + cos (4) ( ) ( ) World Appl. Sci. J., (): 64-69, 0 y(0) =, y (0) = 0 (4) The particular solution of () is; 8 the reason that these initial conditions are sufficiently general to cover all physical systems of interest [0]. Nayfeh [] and Mickens [] had solved the Duffing Then, the general solution of () is; equation with initial conditions (4a,b) using L-P method in second order expansion. According to the standard L-P y( ) = Acos + Bsin method, a new variable; (5) + sin + cos 8 = t, (5) Using the initial conditions (b,c), we obtain the is introduced, where is the frequency of the system that constants A= -/ and B = 0. Thus the solution of (5) is, depends on, that is = ( ). Equation () then becomes y( ) = sin y + y = y, > 0, (6) 8 (6) y(0) =, y (0) = 0 (7a, b) ( cos cos ) + We seek approximate solutions for y and in the form of Note that the solution of y contains a mixed-secular power series in as follows, term, which makes the expansion nonuniform. For a uniform expansion, we cannot permit such secular terms n y(, ) = y 0( ) + y ( ) y n ( ) +... (8) to appear in y, y, y,. The secular term can be eliminated by choosing, n ( ) = n +... (9) = 0 = (7) where 0has been chosen to be unity, (0) = 0=. 8 8 Substituting equation (8) and (9) into equation (6) and (7a,b) and equating of like power of yields the following Thus, without the secular term, the equation (6) system of differential equations for successive becomes, approximations: y ( ) = ( cos cos ) (8) O() : y0 + y0 = 0, (0) According to Nayfeh [], to determine the condition y0 ( 0) = 0, y0( 0) = (0b,c) (7), we do not need to determine the particular solution first as done above. Instead, we only need to inspect the O( ): y + y = y0 y0, (a) inhomogeneous terms in () governing y and choose the coefficient of cos to be zero. y( 0) = y ( 0) = 0 (b,c) Substitute y 0( ) = cos and equation (8) into equation (8), we obtain the first order perturbation O( ) : y + y= y ( + ) y0 y0 y solution of () is, (a) y (, ) = cos + ( cos cos ) + (9) y 0 = y 0 = 0. (b,c) Equation (0a) with initial conditions (0b,c) has a where = t and w( ) = + ( /8) +. periodic solution y 0( ) = cos. Substituting y 0( ) = cos into equation (a) and using cos = ¼(cos + cos) Next, we solve the initial value problem (a,b,c) we obtain the simplified form as, to obtain the solution of second order expansion. The general solution to the resulting differential equation y + y = cos cos () for y (t) is, 65
3 5 ( ) ( ) y = cos 4cos + cos5 04 Therefore, to the second order expansion, the solution to equation () is, (, ) cos ( cos cos ) y = + 5 cos 4cos + 04 cos + t t y( t, ) = cos t t t + 8 cos 4 t t t + 8 cos 4 t 56 + O ( ) World Appl. Sci. J., (): 64-69, 0 (0) The solution of the Duffing equation in fifth order expansion is obtained as follows, () 4 where = t and w( ) = + ( /8) - ( /56) +. However according to Nayfeh [], equation () can be written as, () which is valid when t = O( ). This is the solution of the Duffing equation in second order expansion. Next, we are going to illustrate the solution of the Duffing equation by L-P method using Maple software up to fifth order expansion. The Duffing Equation with Maple: Anyone who uses perturbation methods is struck almost immediately by the amount of algebraic manipulation. As an alternative, we can use a symbolic language like Maple. This mathematical software has a tremendous potential for constructing asymptotic expansions []. In this section, we will illustrate the initial value problem of Duffing equation () with initial conditions (4a,b) by using L-P method with Maple. All commands that we use for solving initial value problems are under the DEtools package in Maple based on Lopez's work [8]. In this work we manage to obtain all the solutions in the fifth order expansions and good agreement with Runge-Kutta Fifth Fourth order (rkf45) numerical solution. y where ( ), cos cos = + + cos 5 5 cos cos cos cos + cos cos5 + cos cos cos cos5 cos cos O 5 ( ) = t () We also present the numerical values of the approximate solutions for () together with the graphs for various values of and several expansions in different order. Table and Fig. show that the solutions do not vary much for values of which are less than However, for = {-0., 0., 0.} show bigger deviation compared to the other values of. Note that, < 0 and > 0 represent pendulum type equations with a hard spring and a soft spring respectively []. Table (a) and Figure (a) shows that at the interval time 0 t 5 does not give any discrepancies for different order of expansion. While Table (b) and Figure (b) show that the higher order expansion gives remarkable accuracy. When t = 99, the relative error of a new result is the smallest compared to the Mickens and Lopez s result. From a quantitative point of view, we can conclude that the higher order approximation is important in giving the accurate result. Other Examples: In this section we will consider the other examples of weakly nonlinear oscillator with no damping. The following equation is a pendulum type equation, 66
4 World Appl. Sci. J., (): 64-69, 0 Table : The approximate solutions of Duffing equation in fifth order expansion for various values of tabulated in the interval 0 b t b t Table (a): The approximate solutions of the Duffing equation at different orders of expansions tabulated in the interval 0 b t b 5 t rkf45 Lopez (00) O( ) Mickens (996) Error O( ) Lopez (00) Error 5 O( ) New results Error Table (b): The approximate solutions of Duffing equation at different orders of expansions tabulated in the interval 95 b t b 00 t rkf45lopez (00) O( ) Mickens (996) Error O( ) Lopez (00) Error 5 O( ) New results Error d y + y = y (5) Both equations will be solved with initial conditions (4a,b). In the literature, Jordan and Smith [0] had solved the problem in the equation (4) and obtained first order expansion using L-P method. Therefore, with the aid of Maple we obtain fifth order expansion as illustrated in the following Fig. (a) and (b). Here, we have shown our present solution is in agreement with the numerical solution by the method of Fehlberg fourth-fifth order Runge-Kutta. Jordan and Smith s result in the interval time 0 t 5 in Fig. (a) and 95 t 00 in Fig. (b) Fig. : The approximate solutions of Duffing equation deviate as time increasing. for various values of, with initial condition Fig. 4(a) and 4(b) show, we present result of the Œ y(0) =, y (0) = 0 at the interval 0 t. equation (5) and it is in agreement with the numerical solution obtained from the maple package. Both solutions d y + y = y (4) indicate that the higher order solution gives more accurate results of the numerical solution compared to the solution Then, we consider the equation, obtained in O( ). 67
5 World Appl. Sci. J., (): 64-69, 0 Fig. (a): The approximate solutions of Duffing equation at different orders of expansions, with initial Fig. (b): The approximate solutions of equation (4) at conditions y(0) =, y (0) = 0 and = 0. in the different orders of expansions, with initial interval 0 t 5 conditions y(0) =, y (0) = 0 and = 0. in the interval 95 t 00 Fig. (b): The approximate solutions of Duffing equation at different orders of expansions, with initial conditions y(0) =, y (0) = 0 and = 0. at the Fig. 4(a): The approximate solutions of equation (5) at interval, 95 t 00 different orders of expansions, with initial conditions y(0) =, y (0) = 0 and = 0. in the interval 0 t 5 Fig. (a): The approximate solutions of equation (4) at Fig. 4(b): The approximate solutions of equation (5) at different orders of expansions, with initial different orders of expansions, with initial conditions y(0) =, y (0) = 0 and = 0. in the conditions y(0) =, y (0) = 0 and = 0. in the interval 0 t 5 interval 95 t 00 68
6 World Appl. Sci. J., (): 64-69, 0 CONCLUSIONS 5. Rand, R.H. and D. Armbruster, 987. Perturbation In this article, we have presented the approximate Methods, Bifurcation Theory and Computer Algebra. NY: Springer. solutions of weakly nonlinear oscillator with no damping 6. nd Heck, A., 996. Introduction to Maple. edition. by considering different initial conditions using L-P New York: Springer-Verlag. method. We implemented L-P method with Maple and 7. Chin, C.M. and A.H. Nayfeh, 999. Perturbation manage to get the periodic solutions in higher order Methods with Maple. Virginia: Dynamic Press. expansions and well-fitted with the numerical solution by 8. Lopez, R., 00. Advanced Engineering Mathematics. rkf 45 method. We also have considered the different Addison Wesley Longman Inc. values of and found that the solutions do not vary 9. Merdan, M., A. Yildirim, A. Gökdogan and much for values of that's less than In future work, S.T. Mohyud-din, 0. Coupling of Homotopy we attempt to find the higher order solution for damping Perturbation, Laplace Transform and Padé oscillatory systems. Approximants for Nonlinear Oscillatory Systems. World Applied Sciences Journal, 6(): 0-8. ACKNOWLEDGEMENTS 0. Jordan, D.W. and P. Smith, 977. Nonlinearity Ordinary Differential Equations. Oxford: Claredon The authors gratefully acknowledge the financial Press. support provided by short term grants, PJP UM grant.. Nayfeh, A.H., 98. Introduction to Perturbation Techniques. New York: John Wiley and Sons. REFERENCES. Mickens, R.E., 996. Oscillations in Planar Dynamic Systems. Series on Advances in Mathematics for. Kevorkian, J. and J.D. Cole, 98. Perturbation Applied Sciences. Singapore: World Scientific Methods in Applied Mathematics. New York: Publishing Co. Pte. Ltd. Springer-Verlag.. Nayfeh, A.H., 99. Introduction to Perturbation. Holmes, M.H., 995. Introduction to Perturbation Techniques. New York: John Wiley and Sons. Methods. New York: Springer-Verlag.. De Jager, E.M. and F. Jiang, 996. The Theory of a Singular Perturbations. Amsterdam: Elsevier. 4. Sanchez, N.E., 996. The Method of Multiple Scales: Asymptotic Solutions and Normal Forms for Nonlinear Oscillatory Problems. Journal of Symbolic Computation, :
Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable
Physica Scripta, Vol. 77, Nº 6, art. 065004 (008) Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable A. Beléndez 1,
More informationKink, singular soliton and periodic solutions to class of nonlinear equations
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 10 Issue 1 (June 015 pp. 1 - Applications and Applied Mathematics: An International Journal (AAM Kink singular soliton and periodic
More informationSolution of an anti-symmetric quadratic nonlinear oscillator by a modified He s homotopy perturbation method
Nonlinear Analysis: Real World Applications, Vol. 10, Nº 1, 416-427 (2009) Solution of an anti-symmetric quadratic nonlinear oscillator by a modified He s homotopy perturbation method A. Beléndez, C. Pascual,
More informationDETERMINATION OF THE FREQUENCY-AMPLITUDE RELATION FOR NONLINEAR OSCILLATORS WITH FRACTIONAL POTENTIAL USING HE S ENERGY BALANCE METHOD
Progress In Electromagnetics Research C, Vol. 5, 21 33, 2008 DETERMINATION OF THE FREQUENCY-AMPLITUDE RELATION FOR NONLINEAR OSCILLATORS WITH FRACTIONAL POTENTIAL USING HE S ENERGY BALANCE METHOD S. S.
More informationApplication of the perturbation iteration method to boundary layer type problems
DOI 10.1186/s40064-016-1859-4 RESEARCH Open Access Application of the perturbation iteration method to boundary layer type problems Mehmet Pakdemirli * *Correspondence: mpak@cbu.edu.tr Applied Mathematics
More informationSolution of Cubic-Quintic Duffing Oscillators using Harmonic Balance Method
Malaysian Journal of Mathematical Sciences 10(S) February: 181 192 (2016) Special Issue: The 3 rd International Conference on Mathematical Applications in Engineering 2014 (ICMAE 14) MALAYSIAN JOURNAL
More informationFeedback Control and Stability of the Van der Pol Equation Subjected to External and Parametric Excitation Forces
International Journal of Applied Engineering Research ISSN 973-456 Volume 3, Number 6 (8) pp. 377-3783 Feedback Control and Stability of the Van der Pol Equation Subjected to External and Parametric Excitation
More informationME DYNAMICAL SYSTEMS SPRING SEMESTER 2009
ME 406 - DYNAMICAL SYSTEMS SPRING SEMESTER 2009 INSTRUCTOR Alfred Clark, Jr., Hopeman 329, x54078; E-mail: clark@me.rochester.edu Office Hours: M T W Th F 1600 1800. COURSE TIME AND PLACE T Th 1400 1515
More information2233. An improved homotopy analysis method with accelerated convergence for nonlinear problems
2233. An improved homotopy analysis method with accelerated convergence for nonlinear problems Hong-wei Li 1, Jun Wang 2, Li-xin Lu 3, Zhi-wei Wang 4 1, 2, 3 Department of Packaging Engineering, Jiangnan
More informationAdditive resonances of a controlled van der Pol-Duffing oscillator
Additive resonances of a controlled van der Pol-Duffing oscillator This paper has been published in Journal of Sound and Vibration vol. 5 issue - 8 pp.-. J.C. Ji N. Zhang Faculty of Engineering University
More informationSolving Zhou Chaotic System Using Fourth-Order Runge-Kutta Method
World Applied Sciences Journal 21 (6): 939-944, 2013 ISSN 11-4952 IDOSI Publications, 2013 DOI: 10.529/idosi.wasj.2013.21.6.2915 Solving Zhou Chaotic System Using Fourth-Order Runge-Kutta Method 1 1 3
More informationSolutions of Nonlinear Oscillators by Iteration Perturbation Method
Inf. Sci. Lett. 3, No. 3, 91-95 2014 91 Information Sciences Letters An International Journal http://dx.doi.org/10.12785/isl/030301 Solutions of Nonlinear Oscillators by Iteration Perturbation Method A.
More informationTHE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A 2 2 SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS
THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS Maria P. Skhosana and Stephan V. Joubert, Tshwane University
More informationFifth-Order Improved Runge-Kutta Method With Reduced Number of Function Evaluations
Australian Journal of Basic and Applied Sciences, 6(3): 9-5, 22 ISSN 99-88 Fifth-Order Improved Runge-Kutta Method With Reduced Number of Function Evaluations Faranak Rabiei, Fudziah Ismail Department
More informationSolution of Excited Non-Linear Oscillators under Damping Effects Using the Modified Differential Transform Method
Article Solution of Excited Non-Linear Oscillators under Damping Effects Using the Modified Differential Transform Method H. M. Abdelhafez Department of Physics and Engineering Mathematics, Faculty of
More informationAPPROXIMATE ANALYTICAL SOLUTIONS TO NONLINEAR OSCILLATIONS OF NON-NATURAL SYSTEMS USING HE S ENERGY BALANCE METHOD
Progress In Electromagnetics Research M, Vol. 5, 43 54, 008 APPROXIMATE ANALYTICAL SOLUTIONS TO NONLINEAR OSCILLATIONS OF NON-NATURAL SYSTEMS USING HE S ENERGY BALANCE METHOD D. D. Ganji, S. Karimpour,
More informationWILEY. Differential Equations with MATLAB (Third Edition) Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg
Differential Equations with MATLAB (Third Edition) Updated for MATLAB 2011b (7.13), Simulink 7.8, and Symbolic Math Toolbox 5.7 Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg All
More informationarxiv: v1 [math.na] 31 Oct 2016
RKFD Methods - a short review Maciej Jaromin November, 206 arxiv:60.09739v [math.na] 3 Oct 206 Abstract In this paper, a recently published method [Hussain, Ismail, Senua, Solving directly special fourthorder
More informationPeriodic Solutions of the Duffing Harmonic Oscillator by He's Energy Balance Method
Journal of pplied and Computational Mechanics, Vol, No 1, (016), 5-1 DOI: 10055/jacm016169 Periodic Solutions of the Duffing Harmonic Oscillator by He's Energy Balance Method M El-Naggar 1, G M Ismail
More informationCubic B-spline Collocation Method for Fourth Order Boundary Value Problems. 1 Introduction
ISSN 1749-3889 print, 1749-3897 online International Journal of Nonlinear Science Vol.142012 No.3,pp.336-344 Cubic B-spline Collocation Method for Fourth Order Boundary Value Problems K.N.S. Kasi Viswanadham,
More informationVariational iteration method for solving multispecies Lotka Volterra equations
Computers and Mathematics with Applications 54 27 93 99 www.elsevier.com/locate/camwa Variational iteration method for solving multispecies Lotka Volterra equations B. Batiha, M.S.M. Noorani, I. Hashim
More informationResearch Article A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
Mathematical Problems in Engineering Volume 1, Article ID 693453, 1 pages doi:11155/1/693453 Research Article A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
More informationDynamics of a mass-spring-pendulum system with vastly different frequencies
Dynamics of a mass-spring-pendulum system with vastly different frequencies Hiba Sheheitli, hs497@cornell.edu Richard H. Rand, rhr2@cornell.edu Cornell University, Ithaca, NY, USA Abstract. We investigate
More informationChapter 14 Three Ways of Treating a Linear Delay Differential Equation
Chapter 14 Three Ways of Treating a Linear Delay Differential Equation Si Mohamed Sah and Richard H. Rand Abstract This work concerns the occurrence of Hopf bifurcations in delay differential equations
More informationAn Exponential Matrix Method for Numerical Solutions of Hantavirus Infection Model
Available at http://pvamu.edu/aam Appl. Appl. Math. ISS: 1932-9466 Vol. 8, Issue 1 (June 2013), pp. 99-115 Applications and Applied Mathematics: An International Journal (AAM) An Exponential Matrix Method
More informationSolution of a Quadratic Non-Linear Oscillator by Elliptic Homotopy Averaging Method
Math. Sci. Lett. 4, No. 3, 313-317 (215) 313 Mathematical Sciences Letters An International Journal http://dx.doi.org/1.12785/msl/4315 Solution of a Quadratic Non-Linear Oscillator by Elliptic Homotopy
More informationAdvanced Mathematical Methods for Scientists and Engineers I
Carl M. Bender Steven A. Orszag Advanced Mathematical Methods for Scientists and Engineers I Asymptotic Methods and Perturbation Theory With 148 Figures Springer CONTENTS! Preface xiii PART I FUNDAMENTALS
More informationEmbedded 5(4) Pair Trigonometrically-Fitted Two Derivative Runge- Kutta Method with FSAL Property for Numerical Solution of Oscillatory Problems
Embedded 5(4) Pair Trigonometrically-Fitted Two Derivative Runge- Kutta Method with FSAL Property for Numerical Solution of Oscillatory Problems N. SENU, N. A. AHMAD, F. ISMAIL & N. BACHOK Institute for
More informationComments on the method of harmonic balance
Comments on the method of harmonic balance Ronald Mickens To cite this version: Ronald Mickens. Comments on the method of harmonic balance. Journal of Sound and Vibration, Elsevier, 1984, 94 (3), pp.456-460.
More informationA Note on Singularly Perturbed System
Jurnal Matematika dan Sains Vol. 7 No. 1, April 00, hal 7-33 A Note on Singularly Perturbed System Hengki Tasman 1), Theo Tuwankotta ), and Wono Setya Budhi 3) Department of Mathematics, ITB E-mails: 1)
More informationAn analytic approach to Van der Pol limit cycle
n analytic approach to Van der Pol limit cycle lberto Strumia Istituto Nazionale di lta Matematica F. Severi (i.n.d.a.m.) - Italy c CRS Registration Number: 6193976149 February 3, 2018 bstract We propose
More informationStudies on Rayleigh Equation
Johan Matheus Tuwankotta Studies on Rayleigh Equation Intisari Dalam makalah ini dipelajari persamaan oscilator tak linear yaitu persamaan Rayleigh : x x = µ ( ( x ) ) x. Masalah kestabilan lokal di sekitar
More informationBritish Journal of Applied Science & Technology 10(2): 1-11, 2015, Article no.bjast ISSN:
British Journal of Applied Science & Technology 10(2): 1-11, 2015, Article no.bjast.18590 ISSN: 2231-0843 SCIENCEDOMAIN international www.sciencedomain.org Solutions of Sequential Conformable Fractional
More informationLecture 17: Ordinary Differential Equation II. First Order (continued)
Lecture 17: Ordinary Differential Equation II. First Order (continued) 1. Key points Maple commands dsolve dsolve[interactive] dsolve(numeric) 2. Linear first order ODE: y' = q(x) - p(x) y In general,
More informationMultiple scale methods
Multiple scale methods G. Pedersen MEK 3100/4100, Spring 2006 March 13, 2006 1 Background Many physical problems involve more than one temporal or spatial scale. One important example is the boundary layer
More informationSolution for the nonlinear relativistic harmonic oscillator via Laplace-Adomian decomposition method
arxiv:1606.03336v1 [math.ca] 27 May 2016 Solution for the nonlinear relativistic harmonic oscillator via Laplace-Adomian decomposition method O. González-Gaxiola a, J. A. Santiago a, J. Ruiz de Chávez
More informationUsing Lagrange Interpolation for Solving Nonlinear Algebraic Equations
International Journal of Theoretical and Applied Mathematics 2016; 2(2): 165-169 http://www.sciencepublishinggroup.com/j/ijtam doi: 10.11648/j.ijtam.20160202.31 ISSN: 2575-5072 (Print); ISSN: 2575-5080
More informationDifferential Equations with Mathematica
Differential Equations with Mathematica THIRD EDITION Martha L. Abell James P. Braselton ELSEVIER ACADEMIC PRESS Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore
More informationAn Exponentially Fitted Non Symmetric Finite Difference Method for Singular Perturbation Problems
An Exponentially Fitted Non Symmetric Finite Difference Method for Singular Perturbation Problems GBSL SOUJANYA National Institute of Technology Department of Mathematics Warangal INDIA gbslsoujanya@gmail.com
More informationColumbus State Community College Mathematics Department. CREDITS: 5 CLASS HOURS PER WEEK: 5 PREREQUISITES: MATH 2173 with a C or higher
Columbus State Community College Mathematics Department Course and Number: MATH 2174 - Linear Algebra and Differential Equations for Engineering CREDITS: 5 CLASS HOURS PER WEEK: 5 PREREQUISITES: MATH 2173
More informationApplication of Homotopy Perturbation Method in Nonlinear Heat Diffusion-Convection-Reaction
0 The Open Mechanics Journal, 007,, 0-5 Application of Homotopy Perturbation Method in Nonlinear Heat Diffusion-Convection-Reaction Equations N. Tolou, D.D. Ganji*, M.J. Hosseini and Z.Z. Ganji Department
More informationFROM EQUILIBRIUM TO CHAOS
FROM EQUILIBRIUM TO CHAOS Practica! Bifurcation and Stability Analysis RÜDIGER SEYDEL Institut für Angewandte Mathematik und Statistik University of Würzburg Würzburg, Federal Republic of Germany ELSEVIER
More informationMaple in Differential Equations
Maple in Differential Equations and Boundary Value Problems by H. Pleym Maple Worksheets Supplementing Edwards and Penney Differential Equations and Boundary Value Problems - Computing and Modeling Preface
More informationA Numerical Solution of Classical Van der Pol-Duffing Oscillator by He s Parameter-Expansion Method
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 15, 709-71 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2013.355 A Numerical Solution of Classical Van der Pol-Duffing Oscillator by
More informationRelative Stability of Mathieu Equation in Third Zone
Relative tability of Mathieu Equation in Third Zone N. Mahmoudian, Ph.D. tudent, Nina.Mahmoudian@ndsu.nodak.edu Tel: 701.231.8303, Fax: 701.231-8913 G. Nakhaie Jazar, Assistant Professor, Reza.N.Jazar@ndsu.nodak.edu
More informationModeling and Experimentation: Compound Pendulum
Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014 Overview This lab focuses on developing a mathematical
More informationENGI 3424 Mid Term Test Solutions Page 1 of 9
ENGI 344 Mid Term Test 07 0 Solutions Page of 9. The displacement st of the free end of a mass spring sstem beond the equilibrium position is governed b the ordinar differential equation d s ds 3t 6 5s
More informationNonlinear Dynamical Systems in Engineering
Nonlinear Dynamical Systems in Engineering . Vasile Marinca Nicolae Herisanu Nonlinear Dynamical Systems in Engineering Some Approximate Approaches Vasile Marinca Politehnica University of Timisoara Department
More informationVariational principles for nonlinear dynamical systems
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 39, NUMBER 2 FEBRUARY 998 Variational principles for nonlinear dynamical systems Vicenç Méndez Grup de Física, Departament de Ciències Ambientals, Facultat de Ciències,
More informationDerivation of amplitude equations for nonlinear oscillators subject to arbitrary forcing
PHYSICAL REVIEW E 69, 066141 (2004) Derivation of amplitude equations for nonlinear oscillators subject to arbitrary forcing Catalina Mayol, Raúl Toral, and Claudio R. Mirasso Department de Física, Universitat
More informationDYNAMICAL SYNTHESIS OF OSCILLATING SYSTEM MASS-SPRING-MAGNET
DYNAMICAL SYNTHESIS OF OSCILLATING SYSTEM MASS-SPRING-MAGNET Rumen NIKOLOV, Todor TODOROV Technical University of Sofia, Bulgaria Abstract. On the basis of a qualitative analysis of nonlinear differential
More informationSYMMETRY REDUCTION AND NUMERICAL SOLUTION OF A THIRD-ORDER ODE FROM THIN FILM FLOW
Mathematical and Computational Applications,Vol. 15, No. 4, pp. 709-719, 2010. c Association for Scientific Research SYMMETRY REDUCTION AND NUMERICAL SOLUTION OF A THIRD-ORDER ODE FROM THIN FILM FLOW E.
More informationCALCULATION OF NONLINEAR VIBRATIONS OF PIECEWISE-LINEAR SYSTEMS USING THE SHOOTING METHOD
Vietnam Journal of Mechanics, VAST, Vol. 34, No. 3 (2012), pp. 157 167 CALCULATION OF NONLINEAR VIBRATIONS OF PIECEWISE-LINEAR SYSTEMS USING THE SHOOTING METHOD Nguyen Van Khang, Hoang Manh Cuong, Nguyen
More informationDifferential Equations
Differential Equations Theory, Technique, and Practice George F. Simmons and Steven G. Krantz Higher Education Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok Bogota
More informationHarmonic balance approach for a degenerate torus of a nonlinear jerk equation
Harmonic balance approach for a degenerate torus of a nonlinear jerk equation Author Gottlieb, Hans Published 2009 Journal Title Journal of Sound and Vibration DOI https://doi.org/10.1016/j.jsv.2008.11.035
More informationRen-He s method for solving dropping shock response of nonlinear packaging system
Chen Advances in Difference Equations 216 216:279 DOI 1.1186/s1662-16-17-z R E S E A R C H Open Access Ren-He s method for solving dropping shock response of nonlinear packaging system An-Jun Chen * *
More informationThree ways of treating a linear delay differential equation
Proceedings of the 5th International Conference on Nonlinear Dynamics ND-KhPI2016 September 27-30, 2016, Kharkov, Ukraine Three ways of treating a linear delay differential equation Si Mohamed Sah 1 *,
More informationSolving nonlinear fractional differential equation using a multi-step Laplace Adomian decomposition method
Annals of the University of Craiova, Mathematics and Computer Science Series Volume 39(2), 2012, Pages 200 210 ISSN: 1223-6934 Solving nonlinear fractional differential equation using a multi-step Laplace
More informationNotes for Expansions/Series and Differential Equations
Notes for Expansions/Series and Differential Equations In the last discussion, we considered perturbation methods for constructing solutions/roots of algebraic equations. Three types of problems were illustrated
More informationDifference Resonances in a controlled van der Pol-Duffing oscillator involving time. delay
Difference Resonances in a controlled van der Pol-Duffing oscillator involving time delay This paper was published in the journal Chaos, Solitions & Fractals, vol.4, no., pp.975-98, Oct 9 J.C. Ji, N. Zhang,
More informationNew Exact Solutions of the Modified Benjamin-Bona-Mahony Equation Yun-jie YANG and Li YAO
06 International Conference on Artificial Intelligence and Computer Science (AICS 06) ISBN: 978--60595-4-0 New Exact Solutions of the Modified Benamin-Bona-Mahony Equation Yun-ie YANG and Li YAO Department
More informationApplied Asymptotic Analysis
Applied Asymptotic Analysis Peter D. Miller Graduate Studies in Mathematics Volume 75 American Mathematical Society Providence, Rhode Island Preface xiii Part 1. Fundamentals Chapter 0. Themes of Asymptotic
More informationSYMBOLIC AND NUMERICAL COMPUTING FOR CHEMICAL KINETIC REACTION SCHEMES
SYMBOLIC AND NUMERICAL COMPUTING FOR CHEMICAL KINETIC REACTION SCHEMES by Mark H. Holmes Yuklun Au J. W. Stayman Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY, 12180 Abstract
More informationEnergy Balance Method for Solving u 1/3 Force Nonlinear Oscillator
Energy Balance Method for Solving u 1/3 Force Nonlinear Oscillator Ebru C. Aslan * & Mustafa Inc 806 Article Department of Mathematics, Firat University, 23119 Elazığ, Türkiye ABSTRACT This paper applies
More informationDIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS
DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS Modern Methods and Applications 2nd Edition International Student Version James R. Brannan Clemson University William E. Boyce Rensselaer Polytechnic
More informationExplicit Solution of Axisymmetric Stagnation. Flow towards a Shrinking Sheet by DTM-Padé
Applied Mathematical Sciences, Vol. 4, 2, no. 53, 267-2632 Explicit Solution of Axisymmetric Stagnation Flow towards a Shrining Sheet by DTM-Padé Mohammad Mehdi Rashidi * Engineering Faculty of Bu-Ali
More informationNON-STATIONARY RESONANCE DYNAMICS OF THE HARMONICALLY FORCED PENDULUM
CYBERNETICS AND PHYSICS, VOL. 5, NO. 3, 016, 91 95 NON-STATIONARY RESONANCE DYNAMICS OF THE HARMONICALLY FORCED PENDULUM Leonid I. Manevitch Polymer and Composite Materials Department N. N. Semenov Institute
More informationLIMITATIONS OF THE METHOD OF MULTIPLE-TIME-SCALES
LIMITATIONS OF THE METHOD OF MULTIPLE-TIME-SCALES Peter B. Kahn Department of Physics and Astronomy State University of New York Stony Brook, NY 794 and Yair Zarmi Department for Energy & Environmental
More informationSOME PROPERTIES OF SOLUTIONS TO POLYNOMIAL SYSTEMS OF DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 2005(2005), No. 40, pp. 7. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) SOME PROPERTIES
More informationENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations
ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential
More informationA Study on Linear and Nonlinear Stiff Problems. Using Single-Term Haar Wavelet Series Technique
Int. Journal of Math. Analysis, Vol. 7, 3, no. 53, 65-636 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ijma.3.3894 A Study on Linear and Nonlinear Stiff Problems Using Single-Term Haar Wavelet Series
More informationWhy are Discrete Maps Sufficient?
Why are Discrete Maps Sufficient? Why do dynamical systems specialists study maps of the form x n+ 1 = f ( xn), (time is discrete) when much of the world around us evolves continuously, and is thus well
More informationApplication of Variational Iteration Method to a General Riccati Equation
International Mathematical Forum,, 007, no. 56, 759-770 Application of Variational Iteration Method to a General Riccati Equation B. Batiha, M. S. M. Noorani and I. Hashim School of Mathematical Sciences
More informationThe Homotopy Perturbation Method for Solving the Modified Korteweg-de Vries Equation
The Homotopy Perturbation Method for Solving the Modified Korteweg-de Vries Equation Ahmet Yildirim Department of Mathematics, Science Faculty, Ege University, 351 Bornova-İzmir, Turkey Reprint requests
More informationA FIELD METHOD FOR SOLVING THE EQUATIONS OF MOTION OF EXCITED SYSTEMS UDC : (045) Ivana Kovačić
FACTA UNIVERSITATIS Series: Mechanics, Automatic Control and Robotics Vol.3, N o,, pp. 53-58 A FIELD METHOD FOR SOLVING THE EQUATIONS OF MOTION OF EXCITED SYSTEMS UDC 534.:57.98(45) Ivana Kovačić Faculty
More informationDifferential Equations and Linear Algebra Supplementary Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University
Differential Equations and Linear Algebra Supplementary Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter 1. Linear algebraic equations 5 1.1. Gaussian elimination
More informationASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS
Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC
More informationThis paper considers the following general nonlinear oscillators [1]:
Progress In Electromagnetics Research M, Vol. 4, 143 154, 2008 HE S ENERGY BALANCE METHOD TO EVALUATE THE EFFECT OF AMPLITUDE ON THE NATURAL FREQUENCY IN NONLINEAR VIBRATION SYSTEMS H. Babazadeh, D. D.
More informationA Mathematica Companion for Differential Equations
iii A Mathematica Companion for Differential Equations Selwyn Hollis PRENTICE HALL, Upper Saddle River, NJ 07458 iv v Contents Preface viii 0. An Introduction to Mathematica 0.1 Getting Started 1 0.2 Functions
More informationMechanical Vibration Analysis Using Maple
Mechanical Vibration Analysis Using Maple WILLIAM F. SÁNCHEZ COSSIO Logic and Computation Group Theoretical and computational Physics Group Physics Engineering Program Sciences and Humanities School EAFIT
More informationInternational Journal of Modern Theoretical Physics, 2012, 1(1): International Journal of Modern Theoretical Physics
International Journal of Modern Theoretical Physics, 2012, 1(1): 13-22 International Journal of Modern Theoretical Physics Journal homepage:www.modernscientificpress.com/journals/ijmtp.aspx ISSN: 2169-7426
More informationWKB and Numerical Compound Matrix Methods for Solving the Problem of Everted Neo-Hookean Spherical Shell Morteza Sanjaranipour 1, Hamed Komeyli 2
Journal of mathematics and computer science (04), 77-90 WKB and Numerical Compound Matrix Methods for Solving the Problem of Everted Neo-Hookean Spherical Shell Morteza Sanjaranipour, Hamed Komeyli Faculty
More informationNONLINEAR NORMAL MODES OF COUPLED SELF-EXCITED OSCILLATORS
NONLINEAR NORMAL MODES OF COUPLED SELF-EXCITED OSCILLATORS Jerzy Warminski 1 1 Department of Applied Mechanics, Lublin University of Technology, Lublin, Poland, j.warminski@pollub.pl Abstract: The main
More informationMathematical Models with Maple
Algebraic Biology 005 151 Mathematical Models with Maple Tetsu YAMAGUCHI Applied System nd Division, Cybernet Systems Co., Ltd., Otsuka -9-3, Bunkyo-ku, Tokyo 11-001, Japan tetsuy@cybernet.co.jp Abstract
More informationA GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-th ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING
Commun. Korean Math. Soc. 26 (2011), No. 4, pp. 695 708 http://dx.doi.org/10.4134/ckms.2011.26.4.695 A GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-th ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION
More informationOn the Solution of the Van der Pol Equation
On the Solution of the Van der Pol Equation arxiv:1308.5965v1 [math-ph] 27 Aug 2013 Mayer Humi Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 0l609
More informationEXACT TRAVELING WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING THE IMPROVED (G /G) EXPANSION METHOD
Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 EXACT TRAVELIN WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USIN THE IMPROVED ( /) EXPANSION METHOD Elsayed M.
More informationSymmetry Reduction of Chazy Equation
Applied Mathematical Sciences, Vol 8, 2014, no 70, 3449-3459 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams201443208 Symmetry Reduction of Chazy Equation Figen AÇIL KİRAZ Department of Mathematics,
More informationDepartment of Engineering Sciences, University of Patras, Patras, Greece
International Differential Equations Volume 010, Article ID 436860, 13 pages doi:10.1155/010/436860 Research Article Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of
More informationA New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution of Oscillatory Problems
Applied Mathematical Sciences, Vol. 1, 16, no. 44, 157-178 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ams.16.64146 A New Embedded Phase-Fitted Modified Runge-Kutta Method for the Numerical Solution
More information1. Consider the initial value problem: find y(t) such that. y = y 2 t, y(0) = 1.
Engineering Mathematics CHEN30101 solutions to sheet 3 1. Consider the initial value problem: find y(t) such that y = y 2 t, y(0) = 1. Take a step size h = 0.1 and verify that the forward Euler approximation
More informationNonlinear dynamics of system oscillations modeled by a forced Van der Pol generalized oscillator
International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-4, Issue-8, August 2017 Nonlinear dynamics of system oscillations modeled by a forced Van der Pol generalized oscillator
More informationMaximum Principles in Differential Equations
Maximum Principles in Differential Equations Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Murray H. Protter Hans F. Weinberger Maximum Principles in Differential
More informationMathematics with Maple
Mathematics with Maple A Comprehensive E-Book Harald Pleym Preface The main objective of these Maple worksheets, organized for use with all Maple versions from Maple 14, is to show how the computer algebra
More informationSolution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type
American ournal of Mathematical Analysis, 015, Vol 3, No 3, 7-84 Available online at http://pubssciepubcom/ajma/3/3/3 Science and Education Publishing DOI:101691/ajma-3-3-3 Solution of System of Linear
More informationarxiv: v4 [cond-mat.other] 14 Apr 2016
arxiv:1502.00545v4 [cond-mat.other] 14 Apr 2016 A simple frequency approximation formula for a class of nonlinear oscillators K. Rapedius, Max-Beckmann-Str.5, D-76227 Karlsruhe, Germany e-mail:kevin.rapedius@gmx.de
More informationComputers, Lies and the Fishing Season
1/47 Computers, Lies and the Fishing Season Liz Arnold May 21, 23 Introduction Computers, lies and the fishing season takes a look at computer software programs. As mathematicians, we depend on computers
More informationSimple conservative, autonomous, second-order chaotic complex variable systems.
Simple conservative, autonomous, second-order chaotic complex variable systems. Delmar Marshall 1 (Physics Department, Amrita Vishwa Vidyapeetham, Clappana P.O., Kollam, Kerala 690-525, India) and J. C.
More information2:2:1 Resonance in the Quasiperiodic Mathieu Equation
Nonlinear Dynamics 31: 367 374, 003. 003 Kluwer Academic Publishers. Printed in the Netherlands. ::1 Resonance in the Quasiperiodic Mathieu Equation RICHARD RAND Department of Theoretical and Applied Mechanics,
More informationHopf Bifurcation Analysis and Approximation of Limit Cycle in Coupled Van Der Pol and Duffing Oscillators
The Open Acoustics Journal 8 9-3 9 Open Access Hopf ifurcation Analysis and Approximation of Limit Cycle in Coupled Van Der Pol and Duffing Oscillators Jianping Cai *a and Jianhe Shen b a Department of
More information